Gradient Stress Field Research Articles

Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams

ABSTRACTIn the present article, wave dispersion behavior of a temperature-dependent functionally graded (FG) nanobeam undergoing rotation subjected to thermal loading is investigated according to nonlocal strain gradient theory, in which the stress enumerates for both nonlocal stress field and the strain gradient stress field. Mori–Tanaka distribution model is considered to express the gradual variation of material properties across the thickness. The governing equations are derived as a function of axial force due to centrifugal stiffening and displacements by applying Hamilton’s principle according to Euler–Bernoulli beam theory. By applying an analytical solution, the dispersion relations of rotating FG nanobeam are obtained by solving an eigenvalue problem. Obviously, numerical results indicate that various parameters such as angular velocity, gradient index, temperature change, wave number, and nonlocality parameter have significant influences on the wave characteristics of rotating FG nanobeams. Hence, the results of this research can provide useful information for the next generation studies and accurate design of nanomachines including nanoscale molecular bearings and nanogears, etc.

Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory

ABSTRACTIn this article, an analytical model for the wave propagation analysis of inhomogeneous functionally graded (FG) nanobeam in thermal environment is developed based on nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. The nanobeam is modeled through a higher order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field supposed to have a nonlinear distribution across the nanobeam thickness. Temperature-dependent material properties of nanobeams are spatially graded based on Mori–Tanaka model. The governing equations of the temperature-dependent functionally graded (FG) nanobeam are derived using the Hamilton’s principle. Numerical examples show that the characteristics of the wave propagation of FG nanobeam are influenced by various parameters such as nonlocality parameter, length scale parameter, gradient index, and temperature changes.

Extended balance of linear momentum from higher gradient stress field theory

AbstractWe present the equation of linear momentum considering higher gradients for stress and body force. Both are approximated via Taylor series expansion within a finite Cauchy cube of dimensions Lc. Whereas linear terms of the series expansion result to the classical balance of linear momentum, terms up to third order yield an extended balance equation. The extension includes an internal length scale L2c caused by surface integrals on the cube. The approach makes use of Cauchy's theorem and standard Newtonian mechanics but constitutive assumptions are not applied. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams

This paper investigates buckling characteristics of a curved functionally graded (FG) nanobeam based on nonlocal strain gradient elasticity theory accounting the stress for not only the nonlocal stress field but also the strain gradients stress field. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of curved FG nanobeam. The governing equations of nonlocal strain gradient curved FG nanobeam in the framework of refined hyperbolic beam model are obtained using Hamilton’s principle and solved implementing an analytical solution for simply-supported and clamped boundary conditions. To validate the present model, the results are compared with those of straight FG nanobeams by extending the radius of nanobeam to infinity. The effects of nonlocal parameter, length scale parameter, power-law exponent, boundary conditions and slenderness ratio on the buckling response of curved FG nanobeams are investigated.

Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory

In this paper, size-dependent free vibration analysis of curved functionally graded nanobeams embedded in Winkler–Pasternak elastic medium is carried out via an analytical solution method. Three kinds of boundary condition namely, simply supported-simply supported, simply supported-clamped and clamped-clamped are investigated. Material properties of curved functionally graded beam change in thickness direction according to the Mori–Tanaka model. Nonlocal strain gradient elasticity theory is adopted to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Nonlocal governing equations of curved functionally graded nanobeam are obtained from Hamilton’s principle based on Euler–Bernoulli beam model. Finally, the influences of length scale parameter, nonlocal parameter, opening angle, elastic medium, material composition, slenderness ratio and boundary conditions on the vibrational characteristics of nanosize curved functionally graded beams are explored.

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

The longitudinal vibration analysis of small-scaled rods is studied in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions for the vibration analysis of small-scaled rods are derived by employing the Hamilton principle. The model contains a nonlocal parameter considering the significance of nonlocal elastic stress field and a material length scale parameter considering the significance of strain gradient stress field. The analytical solutions of predicting the natural frequencies and mode shapes of the rods with some specified boundary conditions are derived. A finite element method is developed and can be used to calculate the vibration problem by arbitrarily applying classical and non-classical boundary conditions. It is shown that the nonlocal strain gradient rod model exerts a stiffness-softening effect when the nonlocal parameter is larger than the material length scale parameter, and exerts a stiffness-hardening effect when the nonlocal parameter is smaller than the material length scale parameter. The higher-order frequencies are more sensitive to the non-classical boundary conditions in comparison with the lower-order frequencies, and the type of non-classical boundary conditions has a little effect on mode shapes.

Free vibration analysis of nonlocal strain gradient beams made of functionally graded material

A size-dependent Timoshenko beam model, which accounts for through-thickness power-law variation of a two-constituent functionally graded (FG) material, is derived in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. The model contains a material length scale parameter introduced to consider the significance of strain gradient stress field and a nonlocal parameter introduced to consider the significance of nonlocal elastic stress field. The influence of through-thickness power-law variation and size-dependent parameters on vibration is investigated. It is found that through-thickness grading of the FG material in the beam has a great effect on the natural frequencies and therefore can be used to control the natural frequencies. The vibration frequencies can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam exerts a stiffness-softening effect. When the material characteristic parameter is larger than the nonlocal parameter, the FG beam exerts a stiffness-hardening effect.

Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory

An analytic model of small-scaled functionally graded (FG) beams for the flexural wave propagation analysis is developed based on the nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. By using the analytic model, the acoustical and optical dispersion relations between phase velocity and wave number are explicitly derived. It is found that an asymptotic phase velocity of both the acoustical and optical branches can be observed. The asymptotic phase velocity can be increased by decreasing the nonlocal parameter or increasing the material characteristic parameter. Furthermore, the power-law index has a significant effect on the acoustical and optical dispersion relations of nano-scaled FG beams. The effects of nonlocal parameter and material characteristic parameter on the acoustical and optical dispersion relation are significant at high wave numbers, however, may be ignored at low wave numbers. The acoustical and optical phase velocities can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter.

Scale effect on migration and coalescence of void in piezoelectric film subjected to gradient stress and electric fields

In this paper, an analytical method is presented to solve the scale effect on migration motion and coalescence of void in piezoelectric solid subjected to gradient stress and electric filed. The piezoelectric solid is considered as an infinite orthotropic piezoelectric film corresponding to the scale of void. The surface diffusion of void in piezoelectric film is taken as the dominant mechanism for the migration motion of the void. Based on the non-local elastic theory considering the interact force among atoms, the influences of small-scale parameter, stress gradient, stress state, electric loading, and shape of void on the migration motion coalescence of void in piezoelectric film are described.

Migration velocity of an elliptical inclusion in piezoelectric film

This paper presents an analytical method to investigate on the migration velocity of an elliptical inclusion in piezoelectric film under coupled gradient stress and electric field. The effects of gradient stress, electric field, the material property and the shape parameter of inclusion on the migration velocity of the elliptical inclusion in piezoelectric material are described and discussed.

AL7N01 Butt Joint Welding Residual Stress Measurement by Ultrasonic Method

Analysis the shortage of ultrasonic stress measurement based on Lcr wave in low spatial resolution. Point out it is inapplicable in measuring gradient stress field like welding residual stress field. In this paper, propagation differential method is proposed to solve the low spatial resolution problem on Lcr wave stress measurement, and its theory is introduced. This method can raise the spatial resolution from tens of millimeter to a several millimeter level with out decline the stress resolution. A welding residual stress field of 7N01 aluminum alloy butted joint plate is measured by this method, and its result is compared with hole drilling method. Source of errors is discussed. It is seen that propagation differential method is applicable in welding residual stress measurement.

Numerical simulation on the thermal deposition of optical surface irradiated by low energy ion beam in ion beam figuring

Based on energy deposition theory of Sigmund, the model of thickness of energy perturbation layer caused by low energy ion bombarding optical surface is built up. Beam current density on the optical surface is obtained by theory analysis, and the model of thermal deposition caused by low energy ion bombarding optical surface is built up. TRIM program is used to simulate the collision between low energy ions and atoms on the optical surface. And then, contributions to the thermal deposition and energy perturbation layer thickness from parameters such as ion energy, ion type and incidence angle are discussed. Finally, by taking the thermal quantity of deposited workpiece as a heat source in ANSYS, temperature field, thermal gradient field and stress field of the workpiece are obtained. The temperature and thermal gradient of the surface radiated by ion beam each present a Gaussian profile, and decrease along the radius from the center to the edge. The stress on the surface is a compressive stress within the radius of Half Maxim, and it is a tension-tension stress from the radius of Half Maxim to the edge.

Motion of an elliptical void in interconnects embedded in matrix under gradient stress field

We present an analytical solution for the motion of an elliptical void in representative interconnects embedded in a matrix with different line aspect (volume) ratio, under gradient stress field. An orthotropic elastic model is used to represent representative interconnects embedded in a matrix. The effects of stress gradient, stress states, an equivalent void size, the orthotropic material characteristic, and the shape parameter of the void on the motion velocity of an elliptical void are described and discussed.

Analytical solution for motion of an elliptical inclusion in gradient stress field

We present a rigorous analytical solution for motion of an elliptical inclusion in isotropic matrix driven by gradient stress field. The interfacial diffusion is considered as the dominant mechanism for the motion. We demonstrate that normal stress gradient on the interface is the major driven force, while the strain energy density gradient is negligible. A key prediction of the solution is that for a given inclusion the motion velocity is proportional to stress gradient only, indicating that the solution is applicable for inclusion motion in nonuniform stress field of varying stress gradient, and that the inclusion tends to move towards the region of lower stress in nonuniform stressed materials.

Suppressed plastic deformation at blunt crack-tips due to strain gradient effects

Large deformation gradients occur near a crack-tip and strain gradient dependent crack-tip deformation and stress fields are expected. Nevertheless, for material length scales much smaller than the scale of the deformation gradients, a conventional elastic–plastic solution is obtained. On the other hand, for significant large material length scales, a conventional elastic solution is obtained. This transition in behaviour is investigated based on a finite strain version of the Fleck–Hutchinson strain gradient plasticity model from 2001. The predictions show that for a wide range of material parameters, the transition from the conventional elastic–plastic to the elastic solution occurs for length scales ranging from 0.001 times the size of the plastic zone to a length scale of the same order of magnitude as the plastic zone.

An analytical solution for motion of an elliptical void under gradient stress field

We present a rigorous analytical solution for motion of an elliptical void under gradient stress field. An elliptical void will move to the region of high stress at a velocity inversely proportional to an equivalent void size, strongly affected by the shape parameter of the void, average stress, stress gradient, and stress states.

Short crack approach for multiaxial fatigue assessment

AbstractA significant part of the fatigue life is spent during short crack growth. Therefore, modelling of short fatigue crack growth offers an opportunity to improve the accuracy of numerical life assessment. Besides stating some general remarks on the short crack approach itself and on multiaxial fatigue criteria, a short crack growth based fatigue life prediction approach for multiaxial non‐proportional loading is presented. This approach accounts for the geometrical size effect by considering the geometry correction functions for semi‐elliptical surface cracks in inhomogeneous gradient stress fields. The geometrical size effect is becoming significant for notch radii smaller than four times the defined technical crack size. Additionally, life influencing factors due to the statistical size effect have been taken into account. The comparison of calculated and experimentally observed fatigue lives of shouldered shafts made of S460N with notch radii of 0.2 to 4.0 mm under non‐proportional tension and torsion loading yields a satisfying accuracy.

Boundary element modeling and analysis of adhesive bonded structural joints

In this paper, a boundary element technique for modeling and analysis of adhesive bonded structural joints is presented. The formulation is developed in the framework of the anisotropic elasticity and attention is focused on the application to composite structural joints built with the splicing concept technique. To model and analyze composite bonded joints a multidomain implementation of the boundary element method has been used. It has been proven well suited and very effective for the characterization of the mechanical behavior of spliced joints, allowing the analysis of the high gradient stress and strain fields near the splice lines as well as the prediction of the overall distribution of the interlaminar tractions. Numerical results show good agreement with analytical solution and finite element analyses.

A fracture mechanics methodology assessment for fretting fatigue

A fracture mechanics methodology was evaluated for a fretting fatigue geometry in which one end of a specimen clamped between fretting pads was loaded in axial fatigue. In previous work, results from experiments on Ti–6Al–4V pads and specimens were evaluated using finite element analyses where stress intensity factors were calculated assuming a single-edge tension, Mode I crack to form. In the present work, mixed-mode behavior was considered and a more realistic crack geometry was incorporated. K I and K II were calculated from stress fields determined from the finite element analysis using a weight function method and assuming a single-edge Mode I/Mode II inclined crack. A correction was then applied based on empirical crack aspect ratio data. K I and K II were analyzed for several experimentally determined combinations of contact pad geometry, specimen thickness, and loading conditions used to obtain a range of normal and shear forces, each corresponding to a fatigue life of 10 7 cycles. The fracture mechanics methodology was used to determine the conditions for propagation or non-propagation of cracks that initiate in the edge of contact region based on a mixed-mode driving force and a short crack corrected threshold. The coefficient of friction was also varied in the analyses. The fracture mechanics approach appears to be a better method for determining the threshold for fretting fatigue than a stress analysis because thresholds for K are better known than criteria for crack initiation in a gradient stress field.

916 圧子圧入法による残留応力測定法の補正値に及ぼす材質および応力場の影響

To widely use "Residual stress measurement for engineering ceramics by indentation fracture method" industrially, the investigation of detailed magnification factor is indispensable. The influences stress distribution on magnification factor and application and magnification factor of this measuement to several ceramic materials were was discussed. The magnification factor which were almost equal to JSMS-SD-4-01 were obtained by considering indent load dependency under stress field of uniformal compression. The magnification factor is almost equivalent in both Si_3N_4 and Al_2O_3. On the other hand, the magnification factor increased with increasing indent load under gradient stress field.